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Hask

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Current revision (20:35, 13 September 2012) (edit) (undo)
(id is a function :))
 
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'''Hask''' refers to a [[Category theory|category]] with types as objects and functions between them as morphisms. However, its use is ambiguous. Sometimes it refers to Haskell (''actual '''Hask'''''), and sometimes it refers to some subset of Haskell where no values are bottom and all functions terminate (''platonic '''Hask'''''). The reason for this is that platonic '''Hask''' has lots of nice properties that actual '''Hask''' does not, and is thus easier to reason in. There is a faithful functor from platonic '''Hask''' to actual '''Hask''' allowing programmers to think in the former to write code in the latter.
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'''Hask''' is the [[Category theory|category]] of Haskell types and functions.
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== '''Hask''' ==
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The objects of '''Hask''' are Haskell types, and the morphisms from objects <hask>A</hask> to <hask>B</hask> are Haskell functions of type <hask>A -> B</hask>. The identity morphism for object <hask>A</hask> is <hask>id :: A -> A</hask>, and the composition of morphisms <hask>f</hask> and <hask>g</hask> is <hask>f . g = \x -> f (g x)</hask>.
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=== Is '''Hask''' even a category? ===
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== Is '''Hask''' even a category? ==
Consider:
Consider:
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</haskell>
</haskell>
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This might be a problem, because <hask>undef1 . id = undef2</hask>. In order to make '''Hask''' a category, we define two functions <hask>f</hask> and <hask>g</hask> as the same morphism if <hask>f a = g a</hask> for all <hask>a</hask>. Thus <hask>undef1</hask> and <hask>undef2</hask> are different ''values'', but the same ''morphism'' in '''Hask'''.
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This might be a problem, because <hask>undef1 . id = undef2</hask>. In order to make '''Hask''' a category, we define two functions <hask>f</hask> and <hask>g</hask> as the same morphism if <hask>f x = g x</hask> for all <hask>x</hask>. Thus <hask>undef1</hask> and <hask>undef2</hask> are different ''values'', but the same ''morphism'' in '''Hask'''.
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=== '''Hask''' is not Cartesian closed ===
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== '''Hask''' is not Cartesian closed ==
Actual '''Hask''' does not have sums, products, or an initial object, and <hask>()</hask> is not a terminal object. The Monad identities fail for almost all instances of the Monad class.
Actual '''Hask''' does not have sums, products, or an initial object, and <hask>()</hask> is not a terminal object. The Monad identities fail for almost all instances of the Monad class.
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! scope="col" | Sum
! scope="col" | Sum
! scope="col" | Product
! scope="col" | Product
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! scope="col" | Product
 +
|-
 +
! scope="row" | Type
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| <hask>data Empty</hask>
 +
| <hask>data () = ()</hask>
 +
| <hask>data Either a b
 +
= Left a | Right b</hask>
 +
| <hask>data (a,b) =
 +
(,) { fst :: a, snd :: b}</hask>
 +
| <hask>data P a b =
 +
P {fstP :: !a, sndP :: !b}</hask>
|-
|-
-
! scope="row" | Definition
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! scope="row" | Requirement
| There is a unique function
| There is a unique function
<br /><hask>u :: Empty -> r</hask>
<br /><hask>u :: Empty -> r</hask>
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<hask>fst . u = f</hask>
<hask>fst . u = f</hask>
<br /><hask>snd . u = g</hask>
<br /><hask>snd . u = g</hask>
 +
| For any functions
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<br /><hask>f :: r -> a</hask>
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<br /><hask>g :: r -> b</hask>
 +
 +
there is a unique function
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<hask>u :: r -> P a b</hask>
 +
 +
such that:
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<hask>fstP . u = f</hask>
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<br /><hask>sndP . u = g</hask>
|-
|-
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! scope="row" | Platonic candidate
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! scope="row" | Candidate
| <hask>u1 r = case r of {}</hask>
| <hask>u1 r = case r of {}</hask>
| <hask>u1 _ = ()</hask>
| <hask>u1 _ = ()</hask>
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<br /><hask>u1 (Right b) = g b</hask>
<br /><hask>u1 (Right b) = g b</hask>
| <hask>u1 r = (f r,g r)</hask>
| <hask>u1 r = (f r,g r)</hask>
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| <hask>u1 r = P (f r) (g r)</hask>
|-
|-
! scope="row" | Example failure condition
! scope="row" | Example failure condition
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| <hask>r ~ ()</hask>
| <hask>r ~ ()</hask>
<br /><hask>f _ = undefined</hask>
<br /><hask>f _ = undefined</hask>
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<br /><hask>g _ = undefined</hask>
 +
| <hask>r ~ ()</hask>
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<br /><hask>f _ = ()</hask>
<br /><hask>g _ = undefined</hask>
<br /><hask>g _ = undefined</hask>
|-
|-
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| <hask>u2 _ = ()</hask>
| <hask>u2 _ = ()</hask>
| <hask>u2 _ = undefined</hask>
| <hask>u2 _ = undefined</hask>
 +
|
|-
|-
! scope="row" | Difference
! scope="row" | Difference
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| <hask>u1 _ = (undefined,undefined)</hask>
| <hask>u1 _ = (undefined,undefined)</hask>
<br /><hask>u2 _ = undefined</hask>
<br /><hask>u2 _ = undefined</hask>
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| <hask>f _ = ()</hask>
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<br /><hask>(fstP . u1) _ = undefined</hask>
|- style="background: red;"
|- style="background: red;"
! scope="row" | Result
! scope="row" | Result
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! scope="col" | FAIL
! scope="col" | FAIL
! scope="col" | FAIL
! scope="col" | FAIL
! scope="col" | FAIL
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|}
|}
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== Platonic '''Hask''' ==
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== "Platonic" '''Hask''' ==
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Because of these difficulties, Haskell developers tend to think in some subset of Haskell where types do not have bottoms. This means that it only includes functions that terminate, and typically only finite values. The corresponding category has the expected initial and terminal objects, sums and products. Instances of Functor and Monad really are endofunctors and monads.
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Because of these difficulties, Haskell developers tend to think in some subset of Haskell where types do not have bottom values. This means that it only includes functions that terminate, and typically only finite values. The corresponding category has the expected initial and terminal objects, sums and products, and instances of Functor and Monad really are endofunctors and monads.
== Links ==
== Links ==

Current revision

Hask is the category of Haskell types and functions.

The objects of Hask are Haskell types, and the morphisms from objects
A
to
B
are Haskell functions of type
A -> B
. The identity morphism for object
A
is
id :: A -> A
, and the composition of morphisms
f
and
g
is
f . g = \x -> f (g x)
.

Contents

1 Is Hask even a category?

Consider: 

undef1 = undefined :: a -> b
undef2 = \_ -> undefined

Note that these are not the same value: 

seq undef1 () = undefined
seq undef2 () = ()
This might be a problem, because
undef1 . id = undef2
. In order to make Hask a category, we define two functions
f
and
g
as the same morphism if
f x = g x
for all
x
. Thus
undef1
and
undef2
are different values, but the same morphism in Hask.

2 Hask is not Cartesian closed

Actual Hask does not have sums, products, or an initial object, and
()
is not a terminal object. The Monad identities fail for almost all instances of the Monad class.
Why Hask isn't as nice as you'd thought.
Initial Object Terminal Object Sum Product Product
Type
data Empty
data () = ()
data Either a b
 = Left a | Right b
data (a,b) =
 (,) { fst :: a, snd :: b}
data P a b =
 P {fstP :: !a, sndP :: !b}
Requirement There is a unique function
u :: Empty -> r
 There is a unique function
u :: r -> ()
 For any functions
f :: a -> r

g :: b -> r

there is a unique function

u :: Either a b -> r

such that:

u . Left = f

u . Right = g
 For any functions
f :: r -> a

g :: r -> b

there is a unique function

u :: r -> (a,b)

such that:

fst . u = f

snd . u = g
 For any functions
f :: r -> a

g :: r -> b

there is a unique function

u :: r -> P a b

such that:

fstP . u = f

sndP . u = g
Candidate
u1 r = case r of {}
u1 _ = ()
u1 (Left a) = f a

u1 (Right b) = g b
u1 r = (f r,g r)
u1 r = P (f r) (g r)
Example failure condition
r ~ ()
r ~ ()
r ~ ()

f _ = ()

g _ = ()
r ~ ()

f _ = undefined

g _ = undefined
r ~ ()

f _ = ()

g _ = undefined
Alternative u
u2 _ = ()
u2 _ = undefined
u2 _ = ()
u2 _ = undefined
Difference
u1 undefined = undefined

u2 undefined = ()
u1 _ = ()

u2 _ = undefined
u1 undefined = undefined

u2 undefined = ()
u1 _ = (undefined,undefined)

u2 _ = undefined
f _ = ()

(fstP . u1) _ = undefined
Result FAIL FAIL FAIL FAIL FAIL

3 "Platonic" Hask

Because of these difficulties, Haskell developers tend to think in some subset of Haskell where types do not have bottom values. This means that it only includes functions that terminate, and typically only finite values. The corresponding category has the expected initial and terminal objects, sums and products, and instances of Functor and Monad really are endofunctors and monads.

4 Links

Retrieved from "http://www.haskell.org/haskellwiki/Hask"

Categories: Mathematics | Theoretical foundations